Abstract

The phenomenon of dynamic symmetry in botanical, the
so-called philotaxian, systems is investigated. A mathematical model of
phylotaxis grounded on Minkovsky's geometry, as well as the original
trigonometric apparatus based upon the golden ratio, are developed. Constant
values of phylotaxis dynamic symmetry are defined. The principle of dynamic
symmetry in architecture, formulated by D. Hambidge, as well as architect Le
Corbusier's Modulor proportional system, are described in terms of philotaxis
mathematics.

The term dynamic symmetry
was for the first time applied by the American architecture researcher J.
Hambidge to a certain principle of proportioning in architecture [11]. Later
this term independently appeared in physics where it was introduced to describe
physical processes that are characterized by invariants [10]. Finally, in the
given research the term dynamic symmetry
is applied to regularity of natural form-shaping that in terms of origin
also appears not to be connectedwith
Hambidge's idea, and, moreover, appearance of this term in physics. However,
all the three variants are deeply interconnected in terms of their meaning
which we are going to show.

At first, we point out
strategic similarity of Hambidge's and our researches. This is a well-known
historical direction which in the field of architecture and art is motivated by
the search for harmony regularities and, thus, is aimed at studying the objects
of nature. Usually architects take interest in the structural regularities of
natural form-shaping and, particularly, in the golden section and Fibonacci
numbers which are regularities standing out by their intriguing role in
architectural form-shaping. It is not accidentally that architects who do
researches so frequently pay attention to botanical phenomenon phyllotaxis which is characterized by
these regularities.

Phyllotaxis was the focus of
attention of J. Hambidge who is the author of the first variant of the dynamic
symmetry concept. While studying this phenomenon, J. Hambidge introduces a law
of the so called uniform growth and
suggests its geometrical interpretation - spiral
of uniform growth or, in other words, - golden spiral(Fig.
1).

Fig. 1. Building the golden spiral according
to Hambidge.

However, the
most important generalization of Hambidge that resulted from his studying the
regularities of natural form-building (phyllotaxis) as well as proportions of
classical architecture comes to the idea of architectural proportioning that is
called dynamic symmetry. J. Hambidge
demonstrates it by means of simple geometrical scheme (Fig. 2).

Fig. 2. Proportional system «Dynamic
symmetry» by J. Hambidge.

This
successive system of rectangles the first of which is a square and the next
successive one is built on the side of the original square equal to 7 and the
diagonal of the previous rectangle. Thus, the series of rectangles is formed
with the ratio of sides expressing the series . In this
series Hambidge differentiates two types of rectangles - static and dynamic.
Static rectangles have their side ratios expressed in integers while dynamic
ones - in irrational numbers. According to Hambidge, dynamic rectangles express
the idea of growth, motion and development. Among such rectangles he
distinguishes three rectangles which long sides equal . However, he
pays particular attention to the rectangle which is directly connected with the «golden
rectangle» Hambidge conducts thorough geometrical
research discovering various manifestations of the golden section in the system
of rectangle . Studying
the geometrical features of this rectangle he illustrates the possibility of
applying it to the analysis of the proportions of classical architecture and
art objects (Fig. 3, 4).

Fig. 3. Three-D model of Parthenon's
proportionsby J. Hambidge.

Fig. 4. Diagramof Greek vase (cantaros) proportionsby J. Hambidge.

Thus, this is in brief
the essence of J. Hambidge's idea of dynamic symmetry. As one can see, it does
not directly derive from the phyllotaxis properties. Generally, J. Hambidge does
not go deep in mathematics of phyllotaxis. In his various schemes illustrating
the regularities of uniform growth or other ideas of proportioning he applies
well-known numerical relations typical of phyllotaxis, including golden
section.

However, his idea of
dynamic symmetry is an original one and in terms of its mathematical content
expresses regularities of quite a general character.It will be possible to demonstrate it after
studying the phyllotaxis research offered below. But before I would like to
predict some possible "unfavourable impressions" that the representatives of
various fields of science may get after reading this text. The author foresees
the possible dissatisfaction on the part of biologists due to schematic layout
and insufficient terminology of biological description of this phenomenon;
mathematicians - who may discover incompliance of symbolics with that used in
phyllotaxis theory; art historians who may be faced with too much mathematical
direction of the research from the viewpoint of art methodology. The author
fully acknowledges all the "troubles" caused by the multidisciplinary character
of the issue.

And still, the author
takes on responsibility for the scientific credibility of the results achieved.
The central task of the research is formulated on the basis of minimum initial
information which is sufficient to grasp the essence of the main issue of the
problem and build the research "from scratch" making it independent of the
research experience accumulated in the field. Though, it is worth mentioning
that this experience is quite substantial. The research of phyllotaxis was in
the centre of attention of many mathematicians and biologists. For
instance,in the second half of the 20th
century it was in the focus of interest of H. Cockster [7], Adler [12], Jean
[13], Schwabe [14], Petukhov [9], etc. who in their works created the modern
"mathematical and biological" theory of phyllotaxis.

The results of the given
research were first published in 1989 [1]. At that time they were new. At any
rate, none of the researchers describing phyllotaxis applied Minkowski's
geometry and the system of hyperbolic trigonometry. We shall briefly describe this
research.

It is known from biology
that mutual arrangement of variousprimordia
that appear on the cones of spindles is characterized by the spiral symmetry.
This principle of situation that was named phyllotaxis
is also clearly observed in dense inflorescences and infructescences, for
instance, on sunflower discs, pine cones and many other types of bioforms (Fig.
5).

On the surfaces of
phyllotaxis forms, dense inflorescences and infructescences in particular, one
can clearly observe left- and right-wound spiral-like rows of structural
elements (primordia, seeds, leaves). The symmetry order of phyllotaxis forms is
usually shown as the ratio of numbers that corresponds to the quantity of left
and right spirals. According to phyllotaxis law these ratios are described by
the combinations of numbers of recurrent rows which accept the rule: u
n = u n - 2 + u n - 1. Widely spread arethe
symmetrical type of phyllotaxis which are described by Fibonacci series: …, 0, 1, 1, 3, 5, 8, 13, 21, 34, … . Numbers of Lucas series…, 1, 3, 4, 7, 11, 18, 29, 34, … are also quite
frequently realized in phyllotaxis, seldom - numbers that belong to the series …,
4,
5,
9,
14,
23,
…
. The order of symmetry in case of Fibonacci phyllotaxis (F-phyllotaxis) is
expressed by the ratios: , , , , , … .

It is typical that in
symmetry cases always there are neighboring numbers of the series. In certain
cases, when on the surfaces of the pattern one can single out three groups of
spirals, the symmetry is shown by means of three numbers. As a rule, the
off-shoots of plants and trees are characterized by the low order of symmetry
while the inflorescences and infructescences - by high symmetry order. For
instance, the order of symmetry with sunflower can be up to , and even .

The interesting indicator
of phyllotaxis structures is the so called D divergence which is the angle of
splitting of twosubsequentprimordia. Divergence
measured in circle fractions in case of F-phyllotaxis is always expressed by
the same number that is the pattern order of symmetry, i.e. it can be equal , , , , , … . As it is known, this series of fractions tends to the limit
» 0,618 of the circle, with which the full
plane angle appears to be divided as to the golden section F.

Some types of phyllotaxis
patterns while growing subsequently change (expand) the order of their
symmetry. It is this peculiarity of phyllotaxis that we name dynamic symmetry. As an example, one can take the sunflower. The sunflower heads
arranged on the various levels of one and the same stem have different
symmetry: the higher the level, i.e. the older the disk, the higher its order
of symmetry is. In symmetry dynamics the following sequence is realized:

… ®®®®® … .

With the change of symmetry the
divergence angle changes respectively. Still on all the disks regardless of the
number of spirals the so called conforming
(angular) characteristicsof spiral
patterns appear to be the same, that is - the spirals cross at a straight
angle.

This
data is enough to identify the objective of the research. It means to
geometrically decipher the phyllotaxis pattern formation process and the key
issue is how the symmetry changes.

At the beginning it was
necessary to do a simple geometrical stylization of phyllotaxis surface and
represent it as a regular flat lattice (Fig. 4). This lattice (Fig. 6d) is
numbered in such a way that the numbers of vertices characterize their distance
from the straight line 00'; at the same time per unit is
accepted the distance to the nearest from 00' point 1. Due to this numbering
rule the order of symmetry of cylindrical lattice in the system of numerical
denomination is expressed in the following way: points adjacent to 0
are numbered 5, 8 and 3 (also -5, -8and -3), i.e. such numbers
that characterize numerical structure of helical spiral of cylindrical lattice.

It is obvious that for
the lattices with different symmetry the numbers of neighboring vertices of O
point will be different.

We shall do the
comparative analysis of the series of lattices illustrating the sequential
stages of symmetry change in phyllotaxis (Fig. 7). According to their metrical
features, all the lattices are the same. Thus, the peculiarity of natural
phyllotaxis lattices to renew their conforming characteristics at any stage of
symmetrical development is taken into account.

It is obvious that for
the lattices with different symmetry the numbers of neighboring vertices of O
point will be different.

Fig.6. Analysis of structural and
numerical features of phyllotaxis lattice:

a -
general look of cedar cone;

b -
scheme of unrolling;

c -
cylindricallattice - idealized form of
cedar cone;

d - unrolling of cylindrical
lattice

We shall do the
comparative analysis of the series of lattices illustrating the sequential
stages of symmetry change in phyllotaxis (Fig. 7). According to their metrical
features, all the lattices are the same. Thus, the peculiarity of natural
phyllotaxis lattices to renew their conforming characteristics at any stage of
symmetrical development is taken into account.

Let us trace the
regularity of transformation of parallelogram 010'. We shall start from the scheme ІІІ. Onecanseethatthesides01and0'ofthisparallelogramareequaltothesections00'1and00'2, which
are the diagonals of this parallelogram at the two previous stages of its
transformation, i.e. at the stages ІandІІ. The
sides of parallelogram 010'4 at the fourth stage appear to be equal the sections 00'2 and 00'3.
This regularity can be traced also at the next stages. Let usprovideseparateillustrationofthisregularity (Fig. 7)

Fig.7. The series of unrollings illustrating sequential stages of cylindrical
lattices symmetrical transformation. All the schemes depict parallelogram OII'O'

The building rule is that
sides of every parallelogram (starting with the third one) are diagonals of two
foregoing parallelograms. As any two successive parallelograms have three
common vertices, it is obvious that all parallelograms have equal square.
Preserving the square is the first interesting feature of parallelogram dynamic
transformation. The second is that straight lines are parallel: parallelogram
at any stage of transformation still is a parallelogram. Here one can draw the
key supposition of the research - preservation of square and parallelism of
straight lines are the features of hyperbolic rotation. It means we deal with
hyperbolic transformation. It is necessary to specify this idea.

Fig.9. Transformation of elementary
parallelogram by means of hyperbolic motion.

Fig.10. Scheme of lattice
transformation by hyperbolic rotation.

.

Figures 9 and 10
illustrate "binding" of the lattice to the scheme of hyperbolic transformation.

Thus, we can state that
hyperbolic rotation is the basis of symmetry transformation illustrated on Fig.
7. This is the key finding of the research that allows to develop an entirely
new look at the issue of phyllotaxis geometry.

First of all, within the
framework of the idea of hyperbolic rotation it is necessary to conduct a
specific analytical and geometric research of a regular lattice. Immediately an
interesting fact is revealed - in metrical characteristics of the lattice
organically present is the value of the golden section. (Fig. 10).

Let us have a look at Fig
11. Here the arrangement of vertices corresponds
to the Fig. 5.9, 5.10 and is characterized by the following conditions:

Let us determine the
abscissa of the point М2, marking = x. From the
condition of symmetry of the points М1 and М2 it goes that = = x- 1 and that the section М1М2 is tilted at an angle of 45о to the axis 0х. It is evident that the difference
of abscissas of the pointsМ1 and М2 is equal to 1. We shall write down
and solve the equation

х-х- 1 = 1,

х2-х- 1 = 0,

х = .

As follows from the
statement of the problem = = F.
Now one can easily make conclusions. The coordinates x and y
of the arbitrary lattice can be expressed in formula:

х = ·Fn,y = ·F- n,(1)

where а- is the semi-axis of hyperbole that
the point under consideration belongs to. Moreover, if for the angular unit (module)
of hyperbolic rotation one takes the angle dividing two neighboring
transformation stages (for instance, stages I and II, II and III, III and IV,
etc. - Fig.7), in other words, the smallest angle, the rotation to which will
result inthe self-alignment of the
lettice, then the exponentn
will coincide with the value of hyperbolic angle (in modules), characterizing
the position of the point under consideration (Fig. 12).

Fig.12. Determining the module of
hyperbolic angle.

In the coordinate system X0Y
the formulae of the coordinates of the arbitrary point look like hyperbolic
functions:

The golden functions
preserve the main features of the "classical" hyperbolic functions and
accommodate with them taking into account the dependence between numbers F
and е (F = e ln F, Fn = e n ln F,). For instance:

= ,Gch n = ch (n ·
ln F);

= ,sh n = Gsh .

Formulae (2) of the
coordinates of the arbitrary point in the system X0Y now can be written
like that:

X = a ·
Gch n,Y = a · Gsh n.(3)

Changes of arbitrary
point coordinates caused by the hyperbolic rotation are expressed by the
formulae:

x' = X · Gch n + Y · Gsh n,

y' = X · Gsh n + Y · Gch n.(4)

In the process of
hyperbolic rotation the lattice is deformed, but periodically, through rotation
module the same metric states are repeated.

Fig. 13 illustrate the
subsequent deformation stages of the lattice elementary triangle which are
fixed through half-module of the rotation. The metric peculiarities of the
triangle are obvious. Fig. 14 show full lattice depictions in two extreme
states. In one of them (Fig. 13) the elementary triangle gets the sizes , , , in the other (Fig. 14), when the lattice is getting square, the
triangle becomes a 45 grad right triangle. These two states set the general
lattice triangle deformation range.

As one can see, the
transformation of the lattice by means of hyperbolic rotation is characterized
by periodicity. Two rotation modules are considered to make a full period (cycle). It is necessary to note that the lattice
state is repeated through one module, however two such states do not coincide,
they are in mirror position as to the hyperboles symmetry axes.

It is important to state:
hyperbolic rotation issymmetry transformation of a regular
lattice. This transformation is not
considered in the classical theory of symmetry.

Fig.13. Characteristic
conditions of lattice elementary triangle.

Fig.14. Two extreme
conditions of the lattice.

It is also necessary to
focus on the following issues. 1)
Unlike the circular rotation when the lattice knots preserve strict mutual
arrangement, the process of hyperbolic rotation leads to the changing of the
mutual arrangement of the knots. 2).
For the symmetric transformation the lattice should be placed onto the
hyperplane in a special way. In general, the basic lattice lines should not
coincide with asymptotes. Fig. 15 (as compared to Fig. 14) illustrates the
effect of such "non-phyllotaxis" transformation. We shall not concentrate on
this important point deserves separate consideration. Let us continue the
research.

Fig.15. Transformation
effect when asymptotes go through the lattice knots.

Following the principle
of lattice numeration the number of any of its points in the coordinate system
where abscissa axis coincides with the direction 00' (Fig. 6, 10), is
numerically equal to its ordinate. We have developed formulae to describe the
coordinates of arbitrary vertex in the system of movable coordinates x'0y'
(Fig. 16).

Fig.16. Determining coordinates of
the arbitrary point in the system of movable coordinates X'oy'.

With any position of the
coordinates the ordinate of the point 1 nearest to 0x' is taken as a unit of
measurement. Thus, for vertices that belong to the hyperbole xy =
1, we have:

x'M = ·
Gch (m -
n),

y'M = ·
Gsh (m + n),(5)

where m- hyperbolic
angle X0M, n- hyperbolic angle X00'.
For vertices that are situated on the hyperbole xy = - 1, the
coordinate formulae will be the following:

x'M' = ·
Gsh (m -
n),

y'M' = ·
Gch (m + n).(6)

Thus, in formulae (5) and
(6) we have got an interpretation of the numbers of those lattice vertices that
slide on the hyperboles nearest to the asymptotes. But the analysis of Fig. 10
shows that the numbers of these vertices are Fibonacci numbers. Taking for the
sake of simplicity m + n = k and keeping in mind that the successive vertices are
situated on the neighboring branches through rotation module, one can come to
the following correspondence:

F1 = ·
Gch 1 = 1,F2 = ·
Gsh 2 = 1,

F3 = ·
Gch 3 = 2,F4 = ·
Gsh 4 = 3,

F5 = ·
Gch 5 = 5,F6 = ·
Gsh 6 = 8,

F7 = ·
Gch 7 = 13,F8 = ·
Gsh 8 = 21,

. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .

Fk = ·
Gch k,Fk + 1 = ·
Gsh (k + 1).(7)

In the lattice system
different recurrent number sequences are realized on different hyperboles.
Introducing the hyperbole scale ratio (g) one gets the generalized variant
of formulae (7):

uk = g ·· Gch k,uk + 1 = g ·· Gsh (k + 1).(8)

There is another way of
representation of the general member uk of recurrent
sequence:

uk = A · Fk + B ·
Fk + 1.(9)

Here A and B
are the so called initial sequence
elements; if recurrent sequence is viewed as an infinite sequence, for
instance, …, 12, -7,
5,
-2,
3,
1,
4,
5,
9,
14,
…
, then initial will be two first numbers of sequence part of fixed signs. In
this case A = 3, B = 1. In Fibonacci sequence it will be A
= 1, B = 0.

There is another way of
obtaining trigonometric interpretation of Fibonacci numbers. There is one
interesting arrangement of coordinate axes - X'0X' that is given by
the directions of square lattice (Fig. 17).

Fig.17. Analysis of
lattice in the system of coordinates
X'OY'.

Here hyperbolic angle X0X'
is equal to half of the module. If the unit of measurement is the side of cell
square, then the coordinates of X' and Y' lattice knots will be
integers. It is evident. For the arbitrary vertex P we have:

X'P = a' ··
Gch (m -),

Y'P = a' ··
Gsh (m + ).(10)

Here a'-
hyperbole radius (the orbits of point P) that coincides with 0X'
axis. If the counting of angle is done with respect to 0X' axis, then formulae
will look like (10):

X'P = a' ··
Gch (Y- 1),

Y'P = a' ··
Gsh Y,(11)

whereY = m + .

This will result in
formulae (7) and (8) that correspond to the special case, in particular, when
the vertices belonging to the hyperbole that is the nearest to the asymptotes,
for whicha' = 1.
In other words, the vertices coordinates belonging to the single-scale
hyperbole are expressed in Fibonacci numbers.

Between the
integral-valued coordinates X' and Y' there is a dependence

X'2 + X' · Y' - Y'2 = a'2.(12)

In fact, it expresses the
hyperbole equation in reference to the coordinates X'0Y'. At the same time,
as the variables X' and Y' successively take on the values
of the adjacent numbers of a certain recurrentseries, the equation (12) illustrates an important feature of recurrent
series, which, taking into account the specificity of formulae (10) and (11),
is recorded in the following way:

|uk2 + uk· uk
+ 1- u2k + 1| = const.(13)

It is obvious that every
series has its own corresponding constant.

Thus, we have got new
mathematical interpretation of numerical properties of phyllotaxis dynamic
symmetry. As we see, symmetry indicators
in phyllotaxis are the integral-valued expressions of golden hyperbolic
functions and dynamics of these indicators is presupposed by the regularity
of integral-valued growth of golden functions. In this case formula (13)
expresses the numerical constant of dynamic symmetry. In particular, for F-phyllotaxis
this constant will be 1:

|Fk2 + Fk· Fk + 1-F2k + 1| = 1.

The phenomenon of
divergence also undergoes mathematical interpretation. At the beginning we
claimed that divergence angle value in case of F-phyllotaxis numerically
coincides with pattern symmetry order, i.e. the divergence angle is changed
together with the change of symmetry. Let us consider Fig. 18. We shall analyze
the deformation character of the elementary parallelogram 011'0' of the flat
lattice. Here there are two important peculiarities. Firstly, in the process of
transformation the parallelogram square is preserved: 0P · 00' = const;
secondly, the deformation bring about the parallel shift of the bases 00'
and 11'.

On the cylinder surface
the elementary parallelogram is transformed into the so called elementary belt, therefore it is clear
that in the process of transformation suchelementary belt will concentrically increase and decrease as to its
height (the square will be left unchanged) and simultaneously there will be
relative angular displacement of its bases that corresponds to the parallel
shift on the involute. Such angular displacement will take place in every
elementary belt and this will stipulate the torsion effect on the cylinder surface in general.

The formula of divergence
angle comes from the ratio which is equal to:

= = .

The transformations
result in:

D = .(14)

Also

=
= -F-
1.(15)

As we see, the angle
limit corresponds to the known value of the so called ,,ideal angle" of F-phyllotaxis.
It is worth noting that in various ,,non-Fibonacci" сases the divergence angle limit has
various values but in all the cases it changes according to the hyperbolic
tangent law.

Fig.18. Analysis of
hyperbolic transformation of cylindrical surface.

In the analysis process
we have also understood the transformation character of the cylinder surface in
the process of symmetrical transformation of the cylinder lattice. It is
possible to state that this transformation results in three interrelated
motions: concentric widening of cylinder along its axis, compression along the
axis and torsion.

We shall specify the
geometric regularity of cylinder transverse circumference transformation. The
radius r of the circumference 00' is increased according to the
hyperbolic cosine law:

r == OO' = .

It is obvious that any
point of the circumference can be simultaneously presented as the end of the
hyperbole movable radius as well as the end of the circumference movable
radius. This peculiarity finds its analytical expression in the following:

With n = 0 we have the
formulae for coordinates' transformation that results from the circular
rotation:

X' = X · cos a + Y · sin a,

Y' = - X · sin a + Y · cos a.

With a = 0 we
have the formulae for coordinates' transformation that results from the
hyperbolic rotation:

X' = X · Gch n + Y ·Gsh n,

Y' = X · Gsh n + Y · Gch n.

These formulae reflect
the complex motion of the point that is simultaneously a circular and
hyperbolic motion. The trajectory of such a motion is a self-intersecting
spiral which can be called a compositeone (Fig.
20), because it actually illustrates the composition of two rotations.

Fig.20. Composite
spiral for which= p.

Thus, the very composite
spiral is an expression of geometrical law of cylinder transverse circumference
transformation. It finds its laconic representation in a formula:

Thus, we have presented
the main findings of the phyllotaxis research that contain mathematical
explanation of the dynamic symmetry of this phenomenon and allow to generalize
on different variants of interpretation of the dynamic symmetry phenomenon,
particularly, in architecture.

However, aiming to fully
present the mathematics of phyllotaxis we shall briefly mention some other
elements of this research. It is necessary to stress that the research was
about cylindrical phyllotaxis. But
this is only a preliminary stage of theoretical idealization of phyllotaxis. Cone is considered to be more adequate
generalization of phyllotaxis pattern. Cylinder as well as disk is considered
to be the special case of cone determined by the extreme values -0 and - of angle of tilt of cone generator
to its axis.

We shall remind the
essence of the existing ideas about geometrical features of phyllotaxis cone
lattices and dynamic mechanism of their pattern formation. It is generally
considered that the structure of phyllotaxis cone lattices (arrangement of the
vertices) is subject to the law of logarithmic spiral [2, 5, 6]. Thus, to the
logarithmic spiral also belong parastichy that form lattices on cone and
disk-like patterns and the so called basic
(or genetic) spiral that
successively runs through all the lattice vertices. Such lattices will be
called logarithmic.

Structural and semantic
character of the lattice is determined by the two indicators qл- ratio of local similitude that can
be found from the ratio qл = , where rkandrk + 1- the distance of two successive
primordia to the cone vertex and D- divergence angle. Here there is
dependence:

qл = q D,(18)

where q- ratio of the basic spiral
similitude.

Fig.21. Analysis of
the general case of logarithmic lattice.

Correspondingly, the
algorithm of logarithmic lattices pattern formation goes to indicating the
primordia motion trajectory, i. e. basic logarithmic spiral and the so called primordial growth intervalDt, or to the intensity of their reproductionN (it is meant that N = ). It is supposed that in naturesuch a process is stipulated due to equal, in terms of time, primordium
growth and thus, the determination conditions for lattice-forming process are
as follows: firstly, invariability of local similitude ratio(qл = = const) and secondly, constant speed of
primordium circular rotation wз:

wз = = const.

So, this is the classical
model of phyllotaxis pattern formation.

The findings of
cylindrical phyllotaxis research result in a principally different idea of cone
lattices modelling. The idea is that the fundamental regularity of the
structural organization of cone phyllotaxis is composite (and not logarythmic) spiral. Parastichy in reality are composite spirals. The same is
genetic spiral. Cone transverse circumference transformation also occurs
according to the law of composite spiral.

Lattices formed according
to the composite spiral law are called natural
lattices (Fig. 22).

Fig.22. Example of
natural lattice with 8:13 symmetry.

They resemble logarithmic
ones but, in fact, they are incompatible with them. The reasons and character
of this incompatibility can be explained on the comparative drawing of
logarithmic and composite spiral (Fig. 23).

Fig.23. Comparative
drawing of logarithmic and composite spiral.

As we see, the
logarithmic spiral is a peculiar asymptote for the composite one. Different
features of logarithmic and natural lattices are explained by different
mathematical nature of these two curves. The natural lattice is characterized
by the initial scale, which is
defined according to the smallest radius of genetic spiral. Evidently, the
notion of initial scale does not have any value in reference to the logarithmic
lattice.

Of principal importance
for the natural lattices is the fact that indicators qл and D in their structure are
not observed in the ideal situations. They only come close to the nominal
values as they withdraw from the centre. In the zone of centre of divergence
from logarithmic regularity they are evident. In fact, this is violation of similitude.
We can specify the divergence regularity. For instance, for qл this regularity is determined by the
change of ratio = , where D- the difference of angular hyperbolic
coordinates of two successive vertices that belong to the base of belt surface.

Before considering a
specific example let us once again stress that the process of lattice pattern
formation is determined by the formula = const. The process parameters are the speedsw, w h and primordium's growth interval Dtor the indicator of their
reproduction intensivenessN = . In every specific case the values ofw, w h and Dt are constant; from them, consequently, one gets nominal characteristics qл and Da = D that determine the symmetry of the lattice restored. So, Da = w·Dt and qл = FDn where Dn = w h·Dt is the angular hyperbolic interval between the successive primordia.

Fig.24. Scheme that
explains the origin of the Golden section in the off-shoots structures.

We have specially chosen
the simplest example which one often observes in nature - on trees and plants
off-shoots. It also explains mathematical reasons for the origin of golden proportion
in the linear correlations of off-shoots structures, in particular, the fact
that proportion discovered in off-shoot natural measurements in fact
approximates golden section. Here golden section cannot be realized in its
absolute value.

And some more words about
natural lattices symmetry transformation. We shall not describe it in details
but emphasize that this transformation is based on composite motion - composite
rotation that combines both hyperbolic and circular rotation. This issue will
be discussed in a separate article.

Finally, we shall do the
generalizations. Besides Hambidge's variant we shall use in our comparative
research another theoretical result which is well-known in architecture -
Modular of French architect of Le Corbusier.

Fig.25. a - scheme of continuous
section division in the golden proportion.

B
- integral-valued scales that were obtained as a result of approximation
ofirrational scales values.

Fig. 25 on the left shows
the scheme suggested by the author of Modular to illustrate the infinite
division of section in the golden proportion. Le Corbusier links such a
division to the proportions of human figure. On the right side of the drawing
one can see the so called red and blue scales where irrational numerical
values obtained by means of infinite division scheme are represented in
approximated integral-valued way. It is clear that irrational numbers express
the golden exponential function. Integral-valued sequences onto which Le
Corbusier transfersthe additive features of the
golden sequence can be considered recurrent to a certain degree of
conventionality. Le Corbusier presents the formulae of red and blue scales in
the following way:

a n = k ∙ Fn, b n = 2k ∙ Fn,

where n is an arbitrary
integer; k- dimension factor equal 1,13 m.

He also accepts that:

a n -
2 + a n - 1 = a n,

b n -
2 + b n - 1 = b n.

Naturally, due to
approximation on certain section of integral-valued sequences additivity is
violated: the golden sequence and its integral-valued approximations ideally
cannot coincide as they express various mathematical regularities.

But we have to show how
to adjust the imbalance. Modular irrational and integral-valued scales are to
be in some ways linked to the hyperbolic coordinates axes (Fig. 26). The sequence
of irrational relations will be reflected on the axis 0y, the integral-valued
sequences will be created from Y' coordinates by the same series of
points located on the neighboring hyperbole branches through one rotation
module.

Fig.26. Reference
of Modular scales toaxes ofhyperbolic coordinates.

So, what are the common
features of phyllotaxis dynamic symmetry, Modular and Hambidge's dynamic
symmetry? The answer is that in all these cases we deal with the same
mathematical regularities, namely, golden hyperbolic functions. Really, on
Hambidge's scheme (Fig. 2) the base of the arbitrary parallelogram can be
interpreted through the golden sine (Gsh x) and diagonal- through the golden cosine (Gch x).
Then we get the formula of J. Hambidge's invariant of dynamic symmetry:

Gch2 x -
Gsh2 x = 1.

Of similar interest is
the possibility of Modular numerical regularities interpretation. The numerical
sequences of red and blue scales are considered recurrent (ignoring some
,,small" ambiguities); it means, we can apply to them the formula of constant
(invariable) of phyllotaxis dynamic symmetry:

|un2 + un· un
- 1- u2
n - 1| = const.

Let us determine the
invariant value for the red scale. We take any couple of neighboring numbers,
for example, 6 and 9. We shall get |62 + 6 ·
9 - 92| = 9. Now we shall determine the
invariant for the blue scale. We take numbers 18 and 30:
|182 + 18 ·
30 - 302| = 6. It is necessary to remember that
the formula used in this case revealscharacteristic dependence(12) of
golden hyperbolic trigonometry, discovered by us.

The numbers of red and
blue scales can be considered as hyperbolic coordinates X' and Y'
of square lattice vertices (see Fig. 17) that belong to two specific
hyperboles. For one of them a' = const = 9, for the other -a' = const = 6.

Our generalization will
be incomplete if we do not remind that features of space-time in special theory
of relativity is also characterized by invariant. Such invariant is the so
called special-temporal interval whichvalue is preserved in the process of transfer from one inertial
system to the other and is expressed in the formulaDt2-Dx2 = const. Here dependence of hyperbolic trigonometry ch2
x -
sh2 x = const also is the mathematical sub base of this
expression

Thus, we have grounds for
the final conclusions.

1.Mathematical research of phyllotaxis described
above establishes the fact that this phenomenon shows regularities of
non-Euclid, to be more specific, pseudo-Euclid geometry more known as
Minkowski's geometry. So far, physics was considered to be the only field where
this geometry was realized. But it was V. E. Vernadskiy who suggested that the
laws of natural pattern formation are based on non-Euclid geometry. [6]. Therefore, our findings can be
considered as those that specify of V. E. Vernadskiy's suggestions.

2.Phyllotaxis geometry and its trigonometric
apparatus reflect peculiarities of mathematics of nature. This is the form of
mathematics where fundamental role belongs to golden section.

3.J. Hambidge's principle of dynamic symmetry, Le
Corbusier's proportional system of Modular,phyllotaxis
dynamic symmetry as well as spatial-temporal invariant of special theory of
relativity have common mathematical grounds of Minkowski's geometry
regularities. Therefore, there are substantial grounds to refer to them as
common regularities of art and nature.